This is an exercise on my study guide for my discrete applications class.

Prove by element argument: A × (B ∩ C) = (A × B) ∩ (A × C)

Now I know that this is the distributive law, but I'm not sure if this proof would work in the exact same way as a union problem would, because I know how to solve that one. Here is my thinking thus far:

Proof: Suppose A, B, and C are sets.

A × (B ∩ C) = (A × B) ∩ (A × C)

Case 1 (a is a member of A): if a belongs to A, then by the definition of the cartesian product, a is also a member of A x B and A x C. By definition of intersection, a belongs to (A × B) ∩ (A × C).

Case 2 (a is a member of B ∩ C): a is a member of both B and C by intersection. a is a member of (A × B) ∩ (A × C) by the definition of intersection.

By definition of a subset, (A × B) ∩ (A × C) is a subset of A × (B ∩ C).